On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space
In a recent paper, Bauschke et al. study ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comono...
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| Veröffentlicht in: | Optimization letters Jg. 16; H. 2; S. 611 - 621 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Berlin, Heidelberg
Springer
01.03.2022
Springer Berlin Heidelberg |
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| ISSN: | 1862-4480, 1862-4472, 1862-4480 |
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| Abstract | In a recent paper, Bauschke et al. study ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent JA.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_A.$$\end{document} In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. zerA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$zer\, A$$\end{document} we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability. |
|---|---|
| AbstractList | In a recent paper, Bauschke et al. study
$$\rho $$
ρ
-comonotonicity as a generalized notion of monotonicity of set-valued operators
A
in Hilbert space and characterize this condition on
A
in terms of the averagedness of its resolvent
$$J_A.$$
J
A
.
In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for
$$\rho $$
ρ
-comonotone operators and obtain an effective rate of metastability. If
A
has a modulus of regularity w.r.t.
$$zer\, A$$
z
e
r
A
we also get a rate of convergence to some zero of
A
even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for
$$\rho $$
ρ
-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability. In a recent paper, Bauschke et al. study ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent JA.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_A.$$\end{document} In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. zerA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$zer\, A$$\end{document} we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability. In a recent paper, Bauschke et al. study ρ -comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent J A . In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for ρ -comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. z e r A we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for ρ -comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability. |
| Author | Kohlenbach, Ulrich |
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| Cites_doi | 10.1007/978-3-319-48311-5 10.1137/0314056 10.1017/S0143385708000011 10.1007/s11856-019-1870-x 10.2168/LMCS-11(4:20)2015 10.1007/s10107-020-01500-6 10.1007/s11856-017-1511-1 10.1137/S0363012903427336 10.1007/978-94-010-1537-0 10.1137/0329022 10.1007/s11856-016-1408-4 10.1016/j.na.2008.01.005 10.1080/02331934.2016.1200577 10.3233/COM-180097 10.1007/s11784-019-0670-6 10.1007/s11590-021-01812-2 10.1142/S0219199717500158 |
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| Keywords | Halpern-type proximal point algorithm Generalized monotone operators Proximal point algorithm Rates of convergence Proof mining Metastability |
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| PublicationTitle | Optimization letters |
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| References | Aoyama, Toyoda (CR1) 2017; 220 Bauschke, Combettes (CR4) 2017 Kohlenbach (CR13) 2021; 28 CR14 Nakajo, Shimoji, Takahashi (CR18) 2007; 8 Kohlenbach (CR11) 2019; 8 Kohlenbach, López-Acedo, Nicolae (CR16) 2019; 232 Barbu (CR3) 1976 Bruck, Reich (CR6) 1977; 3 Rockafellar (CR20) 1976; 14 Bauschke, Moursi, Wang (CR5) 2020 Tao (CR25) 2008; 28 CR2 Takahashi (CR23) 2009; 70 Kohlenbach (CR12) 2020; 9 CR8 Martinet (CR17) 1970; 4 Takahashi (CR22) 2000 CR24 Güler (CR9) 1991; 29 Kohlenbach (CR10) 2016; 216 CR21 Combettes, Pennanen (CR7) 2004; 43 Neumann (CR19) 2015; 11 Kohlenbach, López-Acedo, Nicolae (CR15) 2017; 66 K Aoyama (1738_CR1) 2017; 220 1738_CR14 B Martinet (1738_CR17) 1970; 4 E Neumann (1738_CR19) 2015; 11 PL Combettes (1738_CR7) 2004; 43 U Kohlenbach (1738_CR15) 2017; 66 W Takahashi (1738_CR23) 2009; 70 O Güler (1738_CR9) 1991; 29 U Kohlenbach (1738_CR16) 2019; 232 U Kohlenbach (1738_CR13) 2021; 28 T Tao (1738_CR25) 2008; 28 U Kohlenbach (1738_CR12) 2020; 9 HH Bauschke (1738_CR4) 2017 RE Bruck (1738_CR6) 1977; 3 1738_CR21 1738_CR24 RT Rockafellar (1738_CR20) 1976; 14 W Takahashi (1738_CR22) 2000 V Barbu (1738_CR3) 1976 1738_CR8 U Kohlenbach (1738_CR10) 2016; 216 K Nakajo (1738_CR18) 2007; 8 1738_CR2 HH Bauschke (1738_CR5) 2020 U Kohlenbach (1738_CR11) 2019; 8 |
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| SubjectTerms | Computational Intelligence Generalized monotone operators Halpern-type proximal point algorithm Mathematics Mathematics and Statistics Metastability Numerical and Computational Physics Operations Research/Decision Theory Optimization Original Paper Proof mining Proximal point algorithm Rates of convergence Simulation |
| Title | On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space |
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